On Louchard's Asymptotic Series

2017/10/10 by Hoffman, Michael E.
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1710.03528

Abstract

Recently G. Louchard obtained an asymptotic series ∑_j=0^∞(I_j)/(n^j) for the integral ∫₀¹[xⁿ+(1-x)ⁿ]^\frac1ndx as n→∞, and computed I_j for j≤ 5 in terms of values of the Riemann zeta function. An interesting feature of the computation is that the I_j are first obtained in terms of alternating multiple zeta values, but then everything except products of ordinary zeta values cancels out. We obtain similar formulas for I_n, 6≤ n≤ 9, and conjecture a general formula for I_n in terms of alternating multiple zeta values. We also conjecture that I_n is a rational polynomial in the ordinary zeta values.

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